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Theory · Game Theory

Alexandros Hollender, Gilbert Maystre, Sai Ganesh Nagarajan

Adversarial multiplayer games are an important object of study in multiagent learning. In particular, polymatrix zero-sum games are a multiplayer setting where Nash equilibria are known to be efficiently computable. Towards understanding the limits of tractability in polymatrix games, we study the computation of Nash equilibria in such games where each pair of players plays either a zero-sum or a coordination game. We are particularly interested in the setting where players can be grouped into a small number of teams of identical interest. While the three-team version of the problem is known to be PPAD-complete, the complexity for two teams has remained open. Our main contribution is to prove that the two-team version remains hard, namely it is CLS-hard. Furthermore, we show that this lower bound is tight for the setting where one of the teams consists of multiple independent adversaries. On the way to obtaining our main result, we prove hardness of finding any stationary point in the simplest type of non-convex-concave min-max constrained optimization problem, namely for a class of bilinear polynomial objective functions.

Optimization · Non-Convex

Yikun Bai, Abihith Kothapalli, Hengrong Du, Rocio Diaz Martin, Soheil Kolouri

The Gromov–Wasserstein (GW) problem, a variant of the classical optimal transport (OT) problem, has attracted growing interest in the machine learning and data science communities due to its ability to quantify similarity between measures in different metric spaces. However, like the classical OT problem, GW imposes an equal mass constraint between measures, which restricts its application in many machine learning tasks. To address this limitation, the partial Gromov-Wasserstein (PGW) problem has been introduced. It relaxes the equal mass constraint, allowing the comparison of general positive Radon measures. Despite this, both GW and PGW face significant computational challenges due to their non-convex nature. To overcome these challenges, we propose the linear partial Gromov-Wasserstein (LPGW) embedding, a linearized embedding technique for the PGW problem. For $K$ different metric measure spaces, the pairwise computation of the PGW distance requires solving the PGW problem $\mathcal{O}(K^2)$ times.In contrast, the proposed linearization technique reduces this to $\mathcal{O}(K)$ times. Similar to the linearization technique for the classical OT problem, we prove that LPGW defines a valid metric for metric measure spaces. Finally, we demonstrate the effectiveness of LPGW in practical applications such as shape retrieval and learning with transport-based embeddings, showing that LPGW preserves the advantages of PGW in partial matching while significantly enhancing computational efficiency. The code is available at https://github.com/mint-vu/Linearized_Partial_Gromov_Wasserstein.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Kristian G Georgiev, Roy Rinberg, Sam Park, Shivam Garg, Andrew Ilyas, Aleksander Madry, Seth Neel

Machine unlearning---efficiently removing the effect of a small "forget set" of training data on a pre-trained machine learning model---has recently attracted significant research interest. Despite this interest, however, recent work shows that existing machine unlearning techniques do not hold up to thorough evaluation in non-convex settings. In this work, we introduce a new machine unlearning technique that exhibits strong empirical performance even in such challenging settings. Our starting point is the perspective that the goal of unlearning is to produce a model whose outputs are *statistically indistinguishable* from those of a model re-trained on all but the forget set. This perspective naturally suggests a reduction from the unlearning problem to that of *data attribution, where the goal is to predict the effect of changing the training set on a model's outputs. Thus motivated, we propose the following meta-algorithm, which we call Datamodel Matching (DMM): given a trained model, we (a) use data attribution to *predict* the output of the model if it were re-trained on all but the forget set points; then (b) *fine-tune* the pre-trained model to match these predicted outputs. In a simple convex setting, we show how this approach provably outperforms a variety of iterative unlearning algorithms. Empirically, we use a combination of existing evaluations and a new metric based on the KL-divergence to show that even in non-convex settings, DMM achieves strong unlearning performance relative to existing algorithms. An added benefit of DMM is that it is a meta-algorithm, in the sense that future advances in data attribution translate directly into better unlearning algorithms, pointing to a clear direction for future progress in unlearning.

Deep Learning · Algorithms

Ankit Sonthalia, Alexander Rubinstein, Ehsan Abbasnejad, Seong Joon Oh

It has recently been conjectured that neural network solution sets reachable via stochastic gradient descent (SGD) are convex, considering permutation invariances. This means that a linear path can connect two independent solutions with low loss, given the weights of one of the models are appropriately permuted. However, current methods to test this theory often require very wide networks to succeed. In this work, we conjecture that more generally, the SGD solution set is a star domain that contains a star model that is linearly connected to all the other solutions via paths with low loss values, modulo permutations. We propose the Starlight algorithm that finds a star model of a given learning task. We validate our claim by showing that this star model is linearly connected with other independently found solutions. As an additional benefit of our study, we demonstrate better uncertainty estimates on Bayesian Model Averaging over the obtained star domain. Further, we demonstrate star models as potential substitutes for model ensembles.

Theory · Game Theory

Yanzheng Chen, Jun Yu

Last-iterate convergence behaviours of well-known algorithms are intensively investigated in various games, such as two-player bilinear zero-sum games.However, most known last-iterate convergence properties rely on strict settings where the underlying games must have time-invariant payoffs.Besides, the limited known attempts on the games with time-varying payoffs are in two-player bilinear time-varying zero-sum games and strictly monotone games. By contrast, in other time-varying games, the last-iterate behaviours of two classic algorithms, i.e., extra gradient (EG) and optimistic gradient (OG) algorithms, still lack research, especially the convergence rates in multi-player games.In this paper, we investigate the last-iterate behaviours of EG and OG algorithms for convergent perturbed games, which extend upon the usual model of time-invariant games and incorporate external factors, such as vanishing noises.Using the recently proposed notion of the tangent residual (or its modifications) as the potential function of games and the measure of proximity to the Nash equilibrium, we prove that the last-iterate convergence rates of EG and OG algorithms for perturbed games on bounded convex closed sets are $O({1}/{\sqrt{T}})$ if such games converge to monotone games at rates fast enough and that such a result holds true for certain unconstrained perturbed games. With this result, we address an open questionasking for the last-iterate convergence rate of EG and OG algorithms in constrained and time-varying settings. The above convergence rates are similar to known tight results on corresponding time-invariant games.

Theory · Game Theory

Yanzheng Chen, Jun Yu

Last-iterate convergence behaviours of well-known algorithms are intensively investigated in various games, such as two-player bilinear zero-sum games.However, most known last-iterate convergence properties rely on strict settings where the underlying games must have time-invariant payoffs.Besides, the limited known attempts on the games with time-varying payoffs are in two-player bilinear time-varying zero-sum games and strictly monotone games. By contrast, in other time-varying games, the last-iterate behaviours of two classic algorithms, i.e., extra gradient (EG) and optimistic gradient (OG) algorithms, still lack research, especially the convergence rates in multi-player games.In this paper, we investigate the last-iterate behaviours of EG and OG algorithms for convergent perturbed games, which extend upon the usual model of time-invariant games and incorporate external factors, such as vanishing noises.Using the recently proposed notion of the tangent residual (or its modifications) as the potential function of games and the measure of proximity to the Nash equilibrium, we prove that the last-iterate convergence rates of EG and OG algorithms for perturbed games on bounded convex closed sets are $O({1}/{\sqrt{T}})$ if such games converge to monotone games at rates fast enough and that such a result holds true for certain unconstrained perturbed games. With this result, we address an open questionasking for the last-iterate convergence rate of EG and OG algorithms in constrained and time-varying settings. The above convergence rates are similar to known tight results on corresponding time-invariant games.

Alexandros Hollender, Gilbert Maystre, Sai Ganesh Nagarajan

Adversarial multiplayer games are an important object of study in multiagent learning. In particular, polymatrix zero-sum games are a multiplayer setting where Nash equilibria are known to be efficiently computable. Towards understanding the limits of tractability in polymatrix games, we study the computation of Nash equilibria in such games where each pair of players plays either a zero-sum or a coordination game. We are particularly interested in the setting where players can be grouped into a small number of teams of identical interest. While the three-team version of the problem is known to be PPAD-complete, the complexity for two teams has remained open. Our main contribution is to prove that the two-team version remains hard, namely it is CLS-hard. Furthermore, we show that this lower bound is tight for the setting where one of the teams consists of multiple independent adversaries. On the way to obtaining our main result, we prove hardness of finding any stationary point in the simplest type of non-convex-concave min-max constrained optimization problem, namely for a class of bilinear polynomial objective functions.

Deep Learning · Algorithms

Shireen Kudukkil Manchingal, Muhammad Mubashar, Kaizheng Wang, Keivan Shariatmadar, Fabio Cuzzolin

Machine learning is increasingly deployed in safety-critical domains where erroneous predictions may lead to potentially catastrophic consequences, highlighting the need for learning systems to be aware of how confident they are in their own predictions: in other words, 'to know when they do not know’. In this paper, we propose a novel Random-Set Neural Network (RS-NN) approach to classification which predicts *belief functions* (rather than classical probability vectors) over the class list using the mathematics of *random sets*, i.e., distributions over the collection of *sets* of classes. RS-NN encodes the 'epistemic' uncertainty induced by training sets that are insufficiently representative or limited in size via the size of the convex set of probability vectors associated with a predicted belief function. Our approach outperforms state-of-the-art Bayesian and Ensemble methods in terms of accuracy, uncertainty estimation and out-of-distribution (OoD) detection on multiple benchmarks (CIFAR-10 vs SVHN/Intel-Image, MNIST vs FMNIST/KMNIST, ImageNet vs ImageNet-O). RS-NN also scales up effectively to large-scale architectures (e.g. WideResNet-28-10, VGG16, Inception V3, EfficientNetB2 and ViT-Base-16),exhibits remarkable robustness to adversarial attacks and can provide statistical guarantees in a conformal learning setting.

Applications · Time Series

Akinori F. Ebihara, Taiki Miyagawa, Kazuyuki Sakurai, Hitoshi Imaoka

Time-sensitive machine learning benefits from Sequential Probability Ratio Test (SPRT), which provides an optimal stopping time for early classification of time series. However, in *finite horizon* scenarios, where input lengths are finite, determining the optimal stopping rule becomes computationally intensive due to the need for *backward induction*, limiting practical applicability. We thus introduce FIRMBOUND, an SPRT-based framework that efficiently estimates the solution to backward induction from training data, bridging the gap between optimal stopping theory and real-world deployment. It employs *density ratio estimation* and *convex function learning* to provide statistically consistent estimators for sufficient statistic and conditional expectation, both essential for solving backward induction; consequently, FIRMBOUND minimizes Bayes risk to reach optimality. Additionally, we present a faster alternative using Gaussian process regression, which significantly reduces training time while retaining low deployment overhead, albeit with potential compromise in statistical consistency. Experiments across independent and identically distributed (i.i.d.), non-i.i.d., binary, multiclass, synthetic, and real-world datasets show that FIRMBOUND achieves optimalities in the sense of Bayes risk and speed-accuracy tradeoff. Furthermore, it advances the tradeoff boundary toward optimality when possible and reduces decision-time variance, ensuring reliable decision-making. Code is included in the supplementary materials.

Probabilistic Methods · Everything Else

Kaizheng Wang, Fabio Cuzzolin, Keivan Shariatmadar, David Moens, Hans Hallez

This paper presents an innovative approach, called credal wrapper, to formulating a credal set representation of model averaging for Bayesian neural networks (BNNs) and deep ensembles (DEs), capable of improving uncertainty estimation in classification tasks. Given a finite collection of single predictive distributions derived from BNNs or DEs, the proposed credal wrapper approach extracts an upper and a lower probability bound per class, acknowledging the epistemic uncertainty due to the availability of a limited amount of distributions. Such probability intervals over classes can be mapped on a convex set of probabilities (a credal set) from which, in turn, a unique prediction can be obtained using a transformation called intersection probability transformation. In this article, we conduct extensive experiments on several out-of-distribution (OOD) detection benchmarks, encompassing various dataset pairs (CIFAR10/100 vs SVHN/Tiny-ImageNet, CIFAR10 vs CIFAR10-C, CIFAR100 vs CIFAR100-C and ImageNet vs ImageNet-O) and using different network architectures (such as VGG16, ResNet-18/50, EfficientNet B2, and ViT Base). Compared to the BNN and DE baselines, the proposed credal wrapper method exhibits superior performance in uncertainty estimation and achieves a lower expected calibration error on corrupted data.

Theory · Optimization

Kenshi Abe, Mitsuki Sakamoto, Kaito Ariu, Atsushi Iwasaki

This paper presents a payoff perturbation technique, introducing a strong convexity to players' payoff functions in games. This technique is specifically designed for first-order methods to achieve last-iterate convergence in games where the gradient of the payoff functions is monotone in the strategy profile space, potentially containing additive noise. Although perturbation is known to facilitate the convergence of learning algorithms, the magnitude of perturbation requires careful adjustment to ensure last-iterate convergence. Previous studies have proposed a scheme in which the magnitude is determined by the distance from a periodically re-initialized anchoring or reference strategy. Building upon this, we propose Gradient Ascent with Boosting Payoff Perturbation, which incorporates a novel perturbation into the underlying payoff function, maintaining the periodically re-initializing anchoring strategy scheme. This innovation empowers us to provide faster last-iterate convergence rates against the existing payoff perturbed algorithms, even in the presence of additive noise.

Joao V. Cavalcanti, Laurent Lessard, Ashia Wilson

Backtracking line search is foundational in numerical optimization. The basic idea is to adjust the step size of an algorithm by a {\em constant} factor until some chosen criterion (e.g. Armijo, Descent Lemma) is satisfied. We propose a novel way to adjust step sizes, replacing the constant factor used in regular backtracking with one that takes into account the degree to which the chosen criterion is violated, with no additional computational burden. This light-weight adjustment leads to significantly faster optimization, which we confirm by performing a variety of experiments on over fifteen real world datasets.For convex problems, we prove adaptive backtracking requires no more adjustments to produce a feasible step size than regular backtracking does.For nonconvex smooth problems, we prove adaptive backtracking enjoys the same guarantees of regular backtracking. %same lower bounds that step sizes produced by regular backtracking do.Furthermore, we prove adaptive backtracking preserves the convergence rates of gradient descent and its accelerated variant.

Theory · Learning Theory

Gautam Chandrasekaran, Adam Klivans, Lin Lin Lee, Konstantinos Stavropoulos

We give the first provably efficient algorithms for learning neural networks with respect to distribution shift. We work in the Testable Learning with Distribution Shift framework (TDS learning) of Klivans et al. (2024), where the learner receives labeled examples from a training distribution and unlabeled examples from a test distribution and must either output a hypothesis with low test error or reject if distribution shift is detected. No assumptions are made on the test distribution. All prior work in TDS learning focuses on classification, while here we must handle the setting of nonconvex regression. Our results apply to real-valued networks with arbitrary Lipschitz activations and work whenever the training distribution has strictly sub-exponential tails. For training distributions that are bounded and hypercontractive, we give a fully polynomial-time algorithm for TDS learning one hidden-layer networks with sigmoid activations. We achieve this by importing classical kernel methods into the TDS framework using data-dependent feature maps and a type of kernel matrix that couples samples from both train and test distributions.

Theory · Optimization

Cedar Site Bai, Brian Bullins

In this paper, we provide tight lower bounds for the oracle complexity of minimizing high-order Hölder smooth and uniformly convex functions. Specifically, for a function whose $p^{th}$-order derivatives are Hölder continuous with degree $\nu$ and parameter $H$, and that is uniformly convex with degree $q$ and parameter $\sigma$, we focus on two asymmetric cases: (1) $q > p + \nu$, and (2) $q < p+\nu$. Given up to $p^{th}$-order oracle access, we establish worst-case oracle complexities of $\Omega\left( \left( \frac{H}{\sigma}\right)^\frac{2}{3(p+\nu)-2}\left( \frac{\sigma}{\epsilon}\right)^\frac{2(q-p-\nu)}{q(3(p+\nu)-2)}\right)$ in the first case with an $\ell_\infty$-ball-truncated-Gaussian smoothed hard function and $\Omega\left(\left(\frac{H}{\sigma}\right)^\frac{2}{3(p+\nu)-2}+ \log\log\left(\left(\frac{\sigma^{p+\nu}}{H^q}\right)^\frac{1}{p+\nu-q}\frac{1}{\epsilon}\right)\right)$ in the second case, for reaching an $\epsilon$-approximate solution in terms of the optimality gap. Our analysis generalizes previous lower bounds for functions under first- and second-order smoothness as well as those for uniformly convex functions, and furthermore our results match the corresponding upper bounds in this general setting.

Quanquan Gu, Jinghui Chen, Yuan Cao, Ziyan Yang, Dongruo Zhou

Adaptive gradient methods are workhorses in deep learning. However, the convergence guarantees of adaptive gradient methods for nonconvex optimization have not been thoroughly studied. In this paper, we provide a fine-grained convergence analysis for a general class of adaptive gradient methods including AMSGrad, RMSProp and AdaGrad. For smooth nonconvex functions, we prove that adaptive gradient methods in expectation converge to a first-order stationary point. Our convergence rate is better than existing results for adaptive gradient methods in terms of dimension. In addition, we also prove high probability bounds on the convergence rates of AMSGrad, RMSProp as well as AdaGrad, which have not been established before. Our analyses shed light on better understanding the mechanism behind adaptive gradient methods in optimizing nonconvex objectives.

Mintong Kang, Chejian Xu, Bo Li

Recent advancements in large audio-language models (LALMs) have enabled speech-based user interactions, significantly enhancing user experience and accelerating the deployment of LALMs in real-world applications. However, ensuring the safety of LALMs is crucial to prevent risky outputs that may raise societal concerns or violate AI regulations. Despite the importance of this issue, research on jailbreaking LALMs remains limited due to their recent emergence and the additional technical challenges they present compared to attacks on DNN-based audio models. Specifically, the audio encoders in LALMs, which involve discretization operations, often lead to gradient shattering, hindering the effectiveness of attacks relying on gradient-based optimizations. The behavioral variability of LALMs further complicates the identification of effective (adversarial) optimization targets. Moreover, enforcing stealthiness constraints on adversarial audio waveforms introduces a reduced, non-convex feasible solution space, further intensifying the challenges of the optimization process. To overcome these challenges, we develop AdvWave, the first jailbreak framework against LALMs. We propose a dual-phase optimization method that addresses gradient shattering, enabling effective end-to-end gradient-based optimization. Additionally, we develop an adaptive adversarial target search algorithm that dynamically adjusts the adversarial optimization target based on the response patterns of LALMs for specific queries. To ensure that adversarial audio remains perceptually natural to human listeners, we design a classifier-guided optimization approach that generates adversarial noise resembling common urban sounds. Extensive evaluations on multiple advanced LALMs demonstrate that AdvWave outperforms baseline methods, achieving a 40\% higher average jailbreak attack success rate. Both audio stealthiness metrics and human evaluations confirm that adversarial audio generated by AdvWave is indistinguishable from natural sounds. We believe AdvWave will inspire future research aiming to enhance the safety alignment of LALMs, supporting their responsible deployment in real-world scenarios.

Hongkang Li, Yihua Zhang, shuai ZHANG, Pin-Yu Chen, Sijia Liu, Meng Wang

Task arithmetic refers to editing the pre-trained model by adding a weighted sum of task vectors, each of which is the weight update from the pre-trained model to fine-tuned models for certain tasks. This approach recently gained attention as a computationally efficient inference method for model editing, e.g., multi-task learning, forgetting, and out-of-domain generalization capabilities. However, the theoretical understanding of why task vectors can execute various conceptual operations remains limited, due to the highly non-convexity of training Transformer-based models. To the best of our knowledge, this paper provides the first theoretical characterization of the generalization guarantees of task vector methods on nonlinear Transformers. We consider a conceptual learning setting, where each task is a binary classification problem based on a discriminative pattern. We theoretically prove the effectiveness of task addition in simultaneously learning a set of irrelevant or aligned tasks, as well as the success of task negation in unlearning one task from irrelevant or contradictory tasks. Moreover, we prove the proper selection of linear coefficients for task arithmetic to achieve guaranteed generalization to out-of-domain tasks. All of our theoretical results hold for both dense-weight parameters and their low-rank approximations. Although established in a conceptual setting, our theoretical findings were validated on a practical machine unlearning task using the large language model Phi-1.5 (1.3B).

Deep Learning · Algorithms

Nikos Dimitriadis, Pascal Frossard, François Fleuret

Multi-task trade-offs in machine learning can be addressed via Pareto Front Learning (PFL) methods that parameterize the Pareto Front (PF) with a single model. PFL permits to select the desired operational point during inference, contrary to traditional Multi-Task Learning (MTL) that optimizes for a single trade-off decided prior to training. However, recent PFL methodologies suffer from limited scalability, slow convergence, and excessive memory requirements, while exhibiting inconsistent mappings from preference to objective space. We introduce PaLoRA, a novel parameter-efficient method that addresses these limitations in two ways. First, we augment any neural network architecture with task-specific low-rank adapters and continuously parameterize the Pareto Front in their convex hull. Our approach steers the original model and the adapters towards learning general and task-specific features, respectively. Second, we propose a deterministic sampling schedule of preference vectors that reinforces this division of labor, enabling faster convergence and strengthening the validity of the mapping from preference to objective space throughout training. Our experiments show that PaLoRA outperforms state-of-the-art MTL and PFL baselines across various datasets, scales to large networks, reducing the memory overhead $23.8-31.7$ times compared with competing PFL baselines in scene understanding benchmarks.

Deep Learning · Other Representation Learning

Yingqi Liu, Qinglun Li, Jie Tan, Yifan Shi, Li Shen, Xiaochun Cao

Despite great achievements in algorithm design for Personalized Federated Learning (PFL), research on the theoretical analysis of generalization is still in its early stages. Some theoretical results have investigated the generalization performance of personalized models under the problem setting and hypothesis in convex conditions, which can not reflect the real iteration performance during non-convex training. To further understand the real performance from a generalization perspective, we propose the first algorithm-dependent generalization analysis with uniform stability for the typical PFL method, Partial Model Personalization, on smooth and non-convex objectives. Specifically, we decompose the generalization errors into aggregation errors and fine-tuning errors, then creatively establish a generalization analysis framework corresponding to the gradient estimation process of the personalized training. This framework builds up the bridge among PFL, FL and Pure Local Training for personalized aims in heterogeneous scenarios, which clearly demonstrates the effectiveness of PFL from the generalization perspective. Moreover, we demonstrate the impact of trivial factors like learning steps, stepsizes and communication topologies and obtain the excess risk analysis with optimization errors for PFL. Promising experiments on CIFAR datasets also corroborate our theoretical insights. Our code can be seen in https://github.com/YingqiLiu1999/Understanding-the-Stability-based-Generalization-of-Personalized-Federated-Learning.

Optimization · Non-Convex

Minhak Song, Kwangjun Ahn, Chulhee Yun

Understanding the training dynamics of deep neural networks is challenging due to their high-dimensional nature and intricate loss landscapes. Recent studies have revealed that, along the training trajectory, the gradient approximately aligns with a low-rank top eigenspace of the training loss Hessian, referred to as the dominant subspace. Given this alignment, this paper explores whether neural networks can be trained within the dominant subspace, which, if feasible, could lead to more efficient training methods. Our primary observation is that when the SGD update is projected onto the dominant subspace, the training loss does not decrease further.This suggests that the observed alignment between the gradient and the dominant subspace is spurious. Surprisingly, projecting out the dominant subspace proves to be just as effective as the original update, despite removing the majority of the original update component. We observe similar behavior across practical setups, including the large learning rate regime (also known as Edge of Stability), Sharpness-Aware Minimization, momentum, and adaptive optimizers. We discuss the main causes and implications of this spurious alignment, shedding light on the dynamics of neural network training.